Stochastic Loewner evolution in doubly connected domains

This paper introduces the annulus SLE${}_{\kappa }$ processes in doubly connected domains. Annulus SLE${}_6$ has the same law as stopped radial SLE${}_6$, up to a time-change. For $\kappa ≠6$, some weak equivalence relation exists between annulus SLE${}_{\kappa }$ and radial SLE${}_{\kappa }$. Annulus SLE${}_2$ is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE${}_2$ satisfies the reversibility property. We also consider the disc SLE${}_{\kappa }$ process defined as a limiting case of the annulus SLE’s. Disc SLE${}_6$ has the same law as stopped full plane SLE${}_6$, up to a time-change. Disc SLE${}_2$ is the scaling limit of loop-erased random walk, and is the reversal of radial SLE${}_2$.